From Nadal’s Dodecahedron to Our Icosahedron

Group Members: Francisco, Jasmeet and Parminder

Original Artwork: Desconstrucció d'un dodecàedre by Josep Rey Nadal

Remaking the Original Artwork & The Artist's Vision

To truly understand the intersection of mathematics and sculpture, our first step was to reverse-engineer Josep Rey Nadal’s original work, Desconstrucció d'un dodecàedre.

Nadal describes his work as an exploration of symmetry, spatial bisection, and philosophical balance. By cutting the regular dodecahedron along a continuous path, the solid unzips into two identical, complementary halves that fit together as a three-dimensional manifestation of the Yin and Yang. It visually demonstrates how two opposing, interlocking parts contain the complete whole.

Our Remake: We remade the original dodecahedron using durable cardstock. Replicating the form out of paper required precise measurements and clean folds. Our main snag was managing the structural stability of the paper joints while executing the bisecting cuts. Working through this hands-on assembly revealed how a continuous path along a 3D solid acts as a hidden seam, transforming a flat net into a dynamic, separable 3D network.

Our Artistic and Mathematical Extension: We wanted to build upon Nadal’s concept of bisection while introducing two major changes: a geometric shift and a deliberate material swap. We moved from a dodecahedron (12 pentagonal faces, 20 vertices) to an icosahedron (20 triangular faces, 12 vertices), exploring mathematical duality and symmetry on a different polyhedron. We designed our own 3D replica of the icosahedron (https://faamorim.github.io/hamiltonian-polyhedron/), which provided the digital foundation for our model. 

Nadal’s original piece is crafted from warm, organic wood. For our extension, we transitioned to modern 3D-printed plastic. While wood offers an earthy, traditional feel, 3D printing allowed us to achieve absolute geometric precision and clean, sharp edges. Nadal’s wooden pentagons felt organic and fluid, our 3D-printed triangles introduced sharp energy and structural tension.

Our Interactive Class Activity: To bring our peers into the mathematical and artistic process during our presentation, we wanted an activity that is tactile, low-stakes, and visual. We prepared flat 2D net cutouts of the icosahedron for our classmates so they can experience the puzzle in their own hands. During our upcoming presentation, we will also hand out templates and challenge our classmates to map out a continuous line connecting every single vertex exactly once without lifting their pens or crossing their own paths.

Once they map their paths on paper, we will reveal our 3D-printed icosahedron to demonstrate how their 2D drawings directly translate into the "secret cuts" used to bisect the 3D solid, bridging flat geometry and spatial form.

BC Curriculum Math Connection:

This project serves as a powerful, hands-on entry point for secondary mathematics, aligning closely with the British Columbia Mathematics Curriculum in the following ways:

• Spatial Reasoning & 3D Geometry (Grades 8-10): Students often struggle to visualize how 2D nets fold into 3D objects (and vice versa). This project allows learners to physically deconstruct 3D solids, building spatial awareness and deepening their understanding of vertices, edges, faces, and Euler's characteristic.

Curricular Competencies: It directly supports competencies such as reasoning and analyzing (exploring spatial patterns), understanding and solving (tackling topological puzzles through trial and error), and communicating and representing (connecting mathematical concepts to artistic expression). By rooting abstract geometry in tactile art, it fosters a low-stress environment that encourages inquiry and reduces math anxiety.

A Reflection on Eisner’s Three Curricula

My initial thought when hearing the word curriculum was simply the official document or syllabus, the explicit list of subjects, topics, and learning outcomes that schools advertise and set out to teach. However, Elliot Eisner’s article, "The Three Curricula That All Schools Teach," completely expands this view by demonstrating that schools teach far more, and far less, than their formal documents outline.

Reading Eisner provided three major stops that invited deep reflection:

• The Danger of Extrinsic Rewards: Eisner highlights research showing that relying on extrinsic rewards (like points or early lunch privileges) fosters compliance and turns students into "reward junkies," ultimately dampening their intrinsic interest when rewards are absent.
• Schedules Limit Deep Engagement: Rigid timetables and 50-minute blocks implicitly teach students "not to get too involved in what they do because to become too involved is to court frustration when time runs out".
• The Impact of the Null Curriculum: What schools omit, such as law, economics, the visual arts, or nonverbal, metaphoric modes of thought, actively biases student perspectives and limits the options and criteria used to evaluate human intelligence.

Eisner expands our understanding of curriculum by dividing it into three distinct dimensions: the explicit curriculum (the advertised, formal subjects), the implicit curriculum (the unstated social values, habits, and compliance taught through school culture and structure), and the null curriculum (the options, subjects, and intellectual processes that are left out).

This framework directly connects to mandated provincial frameworks like the BC Curriculum. A mandated provincial document represents the primary explicit curriculum, the public learning outcomes we are accountable for teaching. However, how we deliver that document in the classroom creates an implicit curriculum; if math is taught strictly through rigid schedules, speed drills, and normative grading, students implicitly learn compliance and competition alongside math concepts. Finally, examining what the mandated provincial framework emphasizes,or omits, reveals its null curriculum, reminding us to reflect on whether nonverbal problem-solving, spatial reasoning, or real-world application are being left out. Recognizing all three dimensions allows us to be far more intentional as educators.

Three Stops: A Response to Susan Gerofsky's piece on Math Education in Battleground Schools

My first stop while reading the piece from Battleground Schools was reflecting on the exclusivity of the conservative approach to math education. This perspective feels inherently non-inclusive because it advocates for teaching high-level, abstract technical skills only to a small elite, while leaving just minimal numeric calculation survival skills for the majority. I believe this approach goes against the very nature of learning, which should be inclusive, collaborative, and limitless. Furthermore, the assumption that mathematics is a permanent and infallible discipline is illogical. Math has evolved significantly over the centuries, and a major part of learning is updating our understanding as new facts emerge.

My second stop was realizing how the historical lack of knowledge and expertise among math educators is a massive concern, and likely the root cause of widespread math anxiety among students. It made me pause and think about how poorly taught math, where the focus is on fearfully memorizing formulas and theorems rather than actually understanding them, stays in students' minds forever, often causing them to just give up on the subject. The continuation of this fearful stance toward math from generation to generation is painful to see, and it will only keep getting worse if we don't actively address it as future educators.

My third stop was John Dewey’s analysis of inquiry and experimentation in mathematics. I completely agree with his point that the process of inquiry is messy, complex and uncertain, especially when compared to the traditional, linear delivery of facts to students sitting in rows. However, I believe the only way forward is through that complexity. It is incredibly hard to make math engaging without experimenting and without helping students see the real-world applications of the abstract concepts they are learning. Overall, this was an amazing reading, and the table comparing conservative and progressive views on various aspects of mathematics education was especially clear and helpful for mapping out these differing views.

Favourite and least favourite Math Teachers

It is often said that we usually remember our best and worst teachers and, most of the time, forget about the average ones. This is true to a great extent.

My first memory of having a great math teacher was in Grade 9, when I had a teacher who took teaching us math really seriously and was actually worried about whether we understood it or not. I still remember learning geometry concepts from him, especially the Pythagorean Theorem, and proving how it came into existence. I used to hate math before that, and somehow, his teaching in Grades 9 and 10 helped me develop a keen interest in it. He is probably the major reason I took math even more seriously in my Grade 11 and 12 studies and ended up taking it as a major in college. I am so grateful that I had Mr. S as my math teacher in school, especially when I had a lot of anxiety around math learning.

I did not have any really bad math teachers in particular; I just had a couple of math professors in my undergrad who really did not care if we understood the concepts or not. They would just lecture, dump materials on us, and leave.

As I look back and think about being a math teacher, I realize that an educator can have a huge impact on our life trajectory. My own bittersweet experiences with math educators serve as examples of the kind of math teacher I want, and do not want, to be for my future students. As I prepare to take up the 'mantle of the teacher,' my goal is to act as a gardener in the classroom, preparing the soil for deep, concept-based understanding and providing steady support, rather than simply dumping information and leaving.

1,000 locker problem

When initially presented with the 1,000 locker problem, my immediate strategy was to scale the scenario down to a manageable size. Rather than attempting to conceptualize all 1,000 lockers at once, I mapped out just the first 20 lockers. By manually tracking the open and closed states as the first 20 students passed through, I created a low-stakes environment to test my conjectures and observe emerging patterns.

My primary mathematical thinking relied on inductive reasoning and analyzing factors. After noticing a slight miscalculation in my initial count (I originally thought 15 lockers would be open instead of 16), I paused to consider the core mechanism at play: a locker's state is determined entirely by the number of its factors. Lockers with an even number of factors receive an even number of "touches" and remain open, while those with an odd number of factors receive an odd number of touches and end up closed.

Connecting this observation to the properties of perfect squares, the only integers that possess an odd number of factors because one factor pairs with itself, was a pivotal shift in my approach. It allowed me to simply calculate the largest perfect square under 1,000 ($31*31=961$) to find the final answer.

I have documented my initial scratch work, including the wrong turn I took before realizing my counting error, and photos of these diagrams and factor patterns can be seen attached to my blog post. This was a fun experience and an excellent reminder that we must allow students the space to test small conjectures so their independent mathematical reasoning can firmly take root.

Parminder Kaur

Skemp on two approaches to teaching and learning mathematics

Reading Skemp’s article really made me rethink how we learn math, and a few specific points definitely made me pause. First, his definition of instrumental understanding as just "rules without reasons" hits close to home; it's shocking to realize how often we just want the formula to get the homework done without actually caring about why it works. Second, the town map analogy was a huge eye-opener, it perfectly explains that feeling of being completely stranded on a test the second a question changes even a little bit because you only memorized a fixed path instead of learning the actual layout. Finally, his argument that we're basically teaching two completely different subjects under the single name of "mathematics" was pretty shocking. It makes so much sense that people develop a massive fear of math when they are forced to just memorize meaningless marks on paper rather than seeing it as a connected, living system.

Personally, I strongly side with relational understanding because trying to memorize separate, endless rules for every single problem type sounds exhausting and honestly unsustainable long-term. Even though Skemp is right that instrumental math gives you that quick boost of confidence when you get a page of right answers, it ultimately traps you. Building a mental map of what you're doing takes way more effort upfront, but it's the only way to actually retain the material and feel confident enough to figure things out on your own when a problem doesn't look exactly like the example.

Hello fellow math teachers!

😊